Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking G. 't Hooft Institute for Theoretical Physics, Utrecht, The Netherlands Abstract: A properly defined "naturalness" is imposed on gauge theories. It is an order-of-magnitude restriction that must hold at all energy scales μ. To construct models with complete naturalness for elementary particles, more types of confining gauge theories are needed besides quantum chromodynamics (QCD). We propose a search program for models with improved naturalness and focus on the possibility that currently elementary fermions can be considered composite. Chiral symmetry must then be responsible for the masslessness of these fermions. Thus, we search for QCD-like models where chiral symmetry is not or only partly broken spontaneously. These models are restricted by index relations that often cannot be satisfied by other than unphysical fractional indices. This difficulty has made the author's own search unsuccessful so far. As a by-product, we find yet another reason why in ordinary QCD chiral symmetry must be broken spontaneously. Introduction: The concept of causality requires that macroscopic phenomena follow from microscopic equations. Thus, the properties of liquids and solids follow from the microscopic properties of molecules and atoms. One may either consider these microscopic properties to have been chosen at random by Nature, or attempt to deduce them from even more fundamental equations at smaller length and time scales. In either case, it is unlikely that the microscopic equations contain various free parameters that are carefully adjusted by Nature to give cancelling effects such that the macroscopic systems have some special properties. This is a philosophy we would like to apply to the unified gauge theories: the effective interactions at a large length scale, corresponding to a low energy scale μ₁, should follow from the properties at a much smaller length scale or higher energy scale μ₂, without the requirement that various different parameters at the energy scale μ₂ match with an accuracy of the order of μ₁/μ₂. That would be unnatural. On the other hand, if at the energy scale μ₂ some parameters would be very small, say α(μ₂) = O(μ₁/μ₂), then this may still be natural, provided that this property would not be spoilt by any higher order effects. We now conjecture that the following dogma should be followed: at any energy scale μ, a physical parameter or set of physical parameters α_i(μ) is allowed to be very small only if the replacement α_i(μ) = 0 would increase the symmetry of the system. This is what we mean by naturalness. It is clearly a weaker requirement than that of P. Dirac who insists on having no small numbers at all. It is what one expects if at any mass scale μ > μ₀ some ununderstood theory with strong interactions determines a spectrum of particles with various good or bad symmetry properties. If at μ = μ₀ certain parameters come out to be smallNaturalness, chiral symmetry, and spontaneous chiral symmetry breaking G. 't Hooft Institute for Theoretical Physics, Utrecht, The Netherlands Abstract: A properly defined "naturalness" is imposed on gauge theories. It is an order-of-magnitude restriction that must hold at all energy scales μ. To construct models with complete naturalness for elementary particles, more types of confining gauge theories are needed besides quantum chromodynamics (QCD). We propose a search program for models with improved naturalness and focus on the possibility that currently elementary fermions can be considered composite. Chiral symmetry must then be responsible for the masslessness of these fermions. Thus, we search for QCD-like models where chiral symmetry is not or only partly broken spontaneously. These models are restricted by index relations that often cannot be satisfied by other than unphysical fractional indices. This difficulty has made the author's own search unsuccessful so far. As a by-product, we find yet another reason why in ordinary QCD chiral symmetry must be broken spontaneously. Introduction: The concept of causality requires that macroscopic phenomena follow from microscopic equations. Thus, the properties of liquids and solids follow from the microscopic properties of molecules and atoms. One may either consider these microscopic properties to have been chosen at random by Nature, or attempt to deduce them from even more fundamental equations at smaller length and time scales. In either case, it is unlikely that the microscopic equations contain various free parameters that are carefully adjusted by Nature to give cancelling effects such that the macroscopic systems have some special properties. This is a philosophy we would like to apply to the unified gauge theories: the effective interactions at a large length scale, corresponding to a low energy scale μ₁, should follow from the properties at a much smaller length scale or higher energy scale μ₂, without the requirement that various different parameters at the energy scale μ₂ match with an accuracy of the order of μ₁/μ₂. That would be unnatural. On the other hand, if at the energy scale μ₂ some parameters would be very small, say α(μ₂) = O(μ₁/μ₂), then this may still be natural, provided that this property would not be spoilt by any higher order effects. We now conjecture that the following dogma should be followed: at any energy scale μ, a physical parameter or set of physical parameters α_i(μ) is allowed to be very small only if the replacement α_i(μ) = 0 would increase the symmetry of the system. This is what we mean by naturalness. It is clearly a weaker requirement than that of P. Dirac who insists on having no small numbers at all. It is what one expects if at any mass scale μ > μ₀ some ununderstood theory with strong interactions determines a spectrum of particles with various good or bad symmetry properties. If at μ = μ₀ certain parameters come out to be small